By Theo Bühler
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Example text
Reg] (resp. ) @ Th e category L Con (resp . LPCon) of locally connected (resp. locally pathconnected) topological spaces [and continuous maps ]. (Hint. 1 Special categorical properties of topological constructs Completeness and cocompleteness 1. 1. 1 T h eor em. Let C be a construct. 2. 2. 35 (b) For any set X , any family ((Xi , ~i) )iEl of C- objecis indexed by some class I and any family (Ii : Xi ----+ X)iEl of maps indexed by I there exists a unique C- siruciure C. i) , I i, X , 1) , i. e.
A m onom orphism iff f : X --* Y 'is injective. I b) an epimo rphism iff f : X ---t Y is surjectiv e. P ro of a) ex) Let x , y E X such that f (x) = f(y) · x : (X , ~rI) --* (X, O defined by x (z) = x for each z E X and y : (X, ~rI) ---t ( X ,~) defined y(z) = y for each z E X are C-morphisms (cf. 2 2)) such th at f o x = I 0 y. e. x = y. Sinc e I is injective 1(X' ) = 8(x') for each x' E X' . Thus 1 = 8. (X,~) be Cmorphisms such th a t f 0 1 = f 0 8. 2. 5 Theorem. b) a) (indirect) . Suppose t hat f is not surjective.
JX;))iEI a family of semiuniform convergence spac es, (j; : X ---+ X i)iEI a family of maps, then JX = {F E F(X x X) : (j; x j;)(F) E J X, for each i E I} is th e initial SUConv-structure on X with resp ect to th e given data. e. such t hat A sati sfies the following condit ions: 1) X E A, 2) A E A implies X\A E A , 3) UnEIN An E A whenever (An)nElN is a sequence in At» A map f : (X , A) ---+ (X' , A') between measur abl e spaces is called measurable provided th at f - I[A' ] E A for each A' E A' .
